A008297 -id:A008297 - cp's OEIS frontend

A046089 Triangle read by rows, the Bell transform of (n+2)!/2 without column 0.

1, 3, 1, 12, 9, 1, 60, 75, 18, 1, 360, 660, 255, 30, 1, 2520, 6300, 3465, 645, 45, 1, 20160, 65520, 47880, 12495, 1365, 63, 1, 181440, 740880, 687960, 235305, 35700, 2562, 84, 1, 1814400, 9072000, 10372320, 4452840, 877905, 86940, 4410, 108, 1

Offset: 1

Wolfdieter Lang

Previous name was: A triangle of numbers related to triangle A030523.
a(n,1)= A001710(n+1). a(n,m)=: S1p(3; n,m), a member of a sequence of lower triangular Jabotinsky matrices with nonnegative entries, including S1p(1; n,m)= A008275 (unsigned Stirling first kind), S1p(2; n,m)= A008297(n,m) (unsigned Lah numbers).
Signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A035342(n,m) := S2(3; n,m). The monic row polynomials E(n,x) := sum(a(n,m)*x^m,m=1..n), E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
a(n,m) enumerates unordered increasing n-vertex m-forests composed of m unary trees (out-degree r from {0,1}) whose vertices of depth (distance from the root) j>=1 come in j+2 colors. The k roots (j=0) each come in one (or no) color. - Wolfdieter Lang, Oct 12 2007
a(4,2)=75=4*(3*4)+3*(3*3) from the two types of unordered 2-forests of unary increasing trees associated with the two m=2 parts partitions (1,3) and (2^2) of n=4. The first type has 4 increasing labelings, each coming in (1)*(1*3*4)=12 colored versions, e.g. ((1c1),(2c1,3c3,4c2)) with lcp for vertex label l and color p. Here the vertex labeled 3 has depth j=1, hence 3 colors, c1, c2 and c3, can be chosen and the vertex labeled 4 with j=2 can come in 4 colors, e.g. c1, c2, c3 and c4. Therefore there are 4*(1)*(1*3*4)=48 forests of this (1,3) type. Similarly the (2,2) type yields 3*((1*3)*(1*3))=27 such forests, e.g. ((1c1,3c2)(2c1,4c1)) or ((1c1,3c2)(2c1,4c2)), etc. - Wolfdieter Lang, Oct 12 2007
Also the Bell transform of A001710(n+2) (adding 1,0,0,.. as column 0). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

			Triangle begins:
  [1],
  [3, 1],
  [12, 9, 1],
  [60, 75, 18, 1],
  [360, 660, 255, 30, 1],
  [2520, 6300, 3465, 645, 45, 1],
  ...

Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Wolfdieter Lang, First ten rows.
E. Neuwirth, Recursively defined combinatorial Functions: Extending Galton's board, Discr. Maths. 239 (2001) 33-51.
John Riordan, Letter, Apr 28 1976.

Cf. A001710, A049376, A105752.

Alternating row sums A134138.

a[n_, m_] /; n >= m >= 1 := a[n, m] = (2m + n - 1)*a[n-1, m] + a[n-1, m-1]; a[n_, m_] /; n < m = 0; a[, 0] = 0; a[1, 1] = 1; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]] (* _Jean-François Alcover, Jul 22 2011 *)
a[n_, k_] := -(-1/2)^k*(n+1)!*HypergeometricPFQ[{1-k, n/2+1, (n+3)/2}, {3/2, 2}, 1]/(k-1)!; Table[a[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 28 2013, after Vladimir Kruchinin *)
a[0] = 0; a[n_] := (n + 1)!/2;
T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, a[0]^n], Sum[Binomial[n - 1, j - 1] a[j] T[n - j, k - 1], {j, 0, n - k + 1}]];
Table[T[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 19 2016, after Peter Luschny, updated Jan 01 2021 *)
rows = 9;
a[n_, m_] := BellY[n, m, Table[(k+2)!/2, {k, 0, rows}]];
Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

a(n,k):=(n!*sum((-1)^(k-j)*binomial(k,j)*binomial(n+2*j-1,2*j-1),j,1,k))/(2^k*k!); /* Vladimir Kruchinin, Apr 01 2011 */

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: factorial(n+2)//2, 9) # Peter Luschny, Jan 19 2016

a(n, m) = n!*A030523(n, m)/(m!*2^(n-m)); a(n, m) = (2*m+n-1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1; a(n, m)=0, n

a(n, m) = sum(|S1(n, j)|* A075497(j, m), j=m..n) (matrix product), with S1(n, j) := A008275(n, j) (signed Stirling1 triangle). Priv. comm. to Wolfdieter Lang by E. Neuwirth, Feb 15 2001; see also the 2001 Neuwirth reference.

a(n, k) = (n!*sum(j=1..k, (-1)^(k-j)*binomial(k,j)*binomial(n+2*j-1,2*j-1)))/(2^k*k!) - Vladimir Kruchinin, Apr 01 2011

New name from Peter Luschny, Jan 19 2016

A049352 A triangle of numbers related to triangle A030524.

Original entry on oeis.org

1, 4, 1, 20, 12, 1, 120, 128, 24, 1, 840, 1400, 440, 40, 1, 6720, 16240, 7560, 1120, 60, 1, 60480, 201600, 129640, 27720, 2380, 84, 1, 604800, 2681280, 2275840, 656320, 80080, 4480, 112, 1, 6652800, 38142720, 41370560, 15402240, 2498160, 196560

Offset: 1

Wolfdieter Lang

a(n,1) = A001715(n+2). a(n,m)=: S1p(4; n,m), a member of a sequence of lower triangular Jabotinsky matrices with nonnegative entries including S1p(1; n,m)= A008275 (unsigned Stirling first kind), S1p(2; n,m)= A008297(n,m) (unsigned Lah numbers), S1p(3; n,m)= A046089(n,m).
The signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A035469(n,m) := S2(4; n,m). The monic row polynomials E(n,x) := sum(a(n,m)*x^m,m=1..n), E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
a(n,m) enumerates unordered increasing n-vertex m-forests composed of m unary trees (out-degree r from {0,1}) whose vertices of depth (distance from the root) j>=1 come in j+3 colors. The k roots (j=0) each come in one (or no) color. - Wolfdieter Lang, Oct 12 2007
Also the Bell transform of A001715. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016

			Triangle starts:
{1};
{4,1};
{20,12,1};
{120,128,24,1};
{840,1400,440,40,1};
...
E.g. Row polynomial E(3,x)= 20*x + 12*x^2 + x^3.
a(4,2)=128=4*(4*5)+3*(4*4) from the two types of unordered 2-forests of unary increasing trees associated with the two m=2 parts partitions (1,3) and (2^2) of n=4. The first type has 4 increasing labelings, each coming in (1)*(1*4*5)=20 colored versions, e.g. ((1c1),(2c1,3c4,4c3)) with lcp for vertex label l and color p. Here the vertex labeled 3 has depth j=1, hence 4 colors, c1..c4, can be chosen and the vertex labeled 4 with j=2 can come in 5 colors, e.g. c1..c5. Therefore there are 4*((1)*(1*4*5))=80 forests of this (1,3) type. Similarly the (2,2) type yields 3*((1*4)*(1*4))=48 such forests, e.g. ((1c1,3c2)(2c1,4c4)) or ((1c1,3c3)(2c1,4c2)), etc. - _Wolfdieter Lang_, Oct 12 2007

Wolfdieter Lang, First ten rows.

Cf. A049377 (row sums).

Alternating row sums A134137.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (n+3)!/6, 10); # Peter Luschny, Jan 28 2016

a[n_, k_] := (n!* Sum[(-1)^(k-j)*Binomial[k, j]*Binomial[n+3*j-1, 3*j-1], {j, 1, k}])/(3^k*k!); Table[a[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 26 2013, after Vladimir Kruchinin *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 10;
M = BellMatrix[(# + 3)!/6&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

a(n,k):=(n!*sum((-1)^(k-j)*binomial(k,j)*binomial(n+3*j-1,3*j-1),j,1,k))/(3^k*k!); /* Vladimir Kruchinin, Apr 01 2011 */

a(n, m) = n!*A030524(n, m)/(m!*3^(n-m)); a(n, m) = (3*m+n-1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1; a(n, m)=0, n

a(n,k) = (n!*sum(j=1..k, (-1)^(k-j)*binomial(k,j)*binomial(n+3*j-1,3*j-1)))/(3^k*k!). [Vladimir Kruchinin, Apr 01 2011]

A049403 A triangle of numbers related to triangle A030528; array a(n,m), read by rows (1 <= m <= n).

Original entry on oeis.org

1, 1, 1, 0, 3, 1, 0, 3, 6, 1, 0, 0, 15, 10, 1, 0, 0, 15, 45, 15, 1, 0, 0, 0, 105, 105, 21, 1, 0, 0, 0, 105, 420, 210, 28, 1, 0, 0, 0, 0, 945, 1260, 378, 36, 1, 0, 0, 0, 0, 945, 4725, 3150, 630, 45, 1, 0, 0, 0, 0, 0, 10395, 17325, 6930, 990, 55, 1, 0, 0, 0, 0, 0, 10395, 62370

Offset: 1

Wolfdieter Lang

a(n,1) = A019590(n) = A008279(1,n). a(n,m) =: S1(-1; n,m), a member of a sequence of lower triangular Jabotinsky matrices, including S1(1; n,m) = A008275 (signed Stirling first kind), S1(2; n,m) = A008297(n,m) (signed Lah numbers). a(n,m) matrix is inverse to signed matrix ((-1)^(n-m))*A001497(n-1,m-1) (signed Bessel triangle). The monic row polynomials E(n,x) := Sum_{m=1..n} a(n,m)*x^m, E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).

Exponential Riordan array [1+x, x(1+x/2)]. T(n,k) = A001498(k+1, n-k). - Paul Barry, Jan 15 2009

			Triangle a(n,m) (with rows n >= 1 and columns m >= 1) begins as follows:
  1;                 with row polynomial E(1,x) = x;
  1, 1;              with row polynomial E(2,x) = x^2 + x;
  0, 3,  1;          with row polynomial E(3,x) = 3*x^2 + x^3;
  0, 3,  6,   1;     with row polynomial E(4,x) = 3*x^2 + 6*x^3 + x^4;
  0, 0, 15,  10,   1;
  0, 0, 15,  45,  15,   1;
  0, 0,  0, 105, 105,  21,  1;
  0, 0,  0, 105, 420, 210, 28, 1;
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Wolfdieter Lang, First 10 rows of the array and more.
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 19.

Cf. A000085 (row sums), A001497, A001498, A008275, A008279, A008297, A019590, A030528, A039692.

Variations of this array: A096713, A104556, A122848, A130757.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> `if`(n<2,1,0), 9); # Peter Luschny, Jan 28 2016

t[n_, k_] := k!*Binomial[n, k]/((2 k - n)!*2^(n - k)); Table[ t[n, k], {n, 11}, {k, n}] // Flatten
(* Second program: *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 13;
M = BellMatrix[If[#<2, 1, 0]&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

a(n, m) = n!*A030528(n, m)/(m!*2^(n-m)) for n >= m >= 1.
a(n, m) = (2*m-n+1)*a(n-1, m) + a(n-1, m-1) for n >= m >= 1 with a(n, m) = 0 for n < m, a(n, 0) := 0, and a(1, 1) = 1. [The 0th column does not appear in this array. - Petros Hadjicostas, Oct 28 2019]
E.g.f. for the m-th column: (x*(1 + x/2))^m/m!.
a(n,m) = A122848(n,m). - R. J. Mathar, Jan 14 2011

A052871 Expansion of e.g.f. -LambertW(x/(-1+x)).

Original entry on oeis.org

0, 1, 4, 27, 268, 3585, 60846, 1255471, 30535912, 855688833, 27148954330, 962037575631, 37659124454700, 1613921425656865, 75156944627712598, 3778932799275876495, 204039148080188427856, 11774630933193827543553

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Previous name was: A simple grammar.

G. C. Greubel, Table of n, a(n) for n = 0..369
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 842
Index entries for reversions of series

Cf. A008297, A060356.

spec := [S,{C=Sequence(Z,1 <= card),B=Set(S),S=Prod(B,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

CoefficientList[Series[-LambertW[x/(-1+x)], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)

makelist(sum((n!/k!)*binomial(n-1, k-1)*k^(k-1), k, 1, n), n, 0, 17);  /* Bruno Berselli, May 25 2011 */

x='x+O('x^50); concat([0], Vec(serlaplace(- lambertw(x/(-1+x))) )) \\ G. C. Greubel, Nov 08 2017

E.g.f.: -LambertW(x/(-1+x))
a(n) = Sum_{k=1..n} (n!/k!)*binomial(n-1, k-1)*k^(k-1). - Vladeta Jovovic, Sep 17 2003
a(n) ~ (1+exp(-1))^(n+1/2)*n^(n-1). - Vaclav Kotesovec, Sep 30 2013
From Seiichi Manyama, Sep 10 2024: (Start)
E.g.f. A(x) satisfies A(x) = x * (A(x) + exp(A(x))).
E.g.f.: Series_Reversion( x / (x + exp(x)) ). (End)

New name using e.g.f., Vaclav Kotesovec, Sep 30 2013

A187536 Partial sums of the central Lah numbers (A187535).

Original entry on oeis.org

1, 3, 39, 1239, 60039, 3870279, 311229639, 29993362119, 3369233266119, 432276047602119, 62366420037720519, 9994350965362162119, 1761334292457572030919, 338557476887113316030919, 70488382605888266852030919, 15802755831536546966525630919

Offset: 0

Emanuele Munarini, Mar 11 2011

Cf. A187535, A008297, A111596, A187538 - A187540, A187542 - A187548.

A187536 := proc(n) add(A187535(i),i=0..n) ; end proc:
seq(A187536(n),n=0..10) ; # R. J. Mathar, Mar 20 2011

Table[1 + Sum[Binomial[2k-1,k-1](2k)!/k!, {k, 1, n}], {n, 0, 20}]

makelist(1+sum(binomial(2*k-1,k-1)*(2*k)!/k!,k,1,n),n,0,12);

a(n) = 1 + Sum_{k=0..n} binomial(2k-1,k-1)*(2k)!/k!.
(n+2)*a(n+2) - (16n^2 + 49n +3 8)*a(n+1) + 4 *(2n+3)^2*a(n) = 0.
Asymptotically a(n) ~ 2^(4n)n^n exp(-n)/sqrt(2n*pi).

A187538 Alternating partial sums of the central Lah numbers (A187535).

Original entry on oeis.org

1, 1, 35, 1165, 57635, 3752605, 303606755, 29378525725, 3309861378275, 425596952957725, 61508547037160675, 9870475998287280925, 1741469465493922587875, 335054673129161821412125, 69814770455871991714587875, 15662452678474786707959012125, 3764014801927115965888623387875

Offset: 0

Emanuele Munarini, Mar 11 2011

Cf. A187536, A008297, A111596, A187536, A187539, A187540, A187542 - A187548.

A187538 := proc(n) add( (-1)^(n+k)*A187535(k),k=0..n) ; end proc:
seq(A187538(n),n=0..10) ; # R. J. Mathar, Mar 21 2011

Table[(-1)^n + Sum[(-1)^(n-k)Binomial[2k-1,k-1](2k)!/k!, {k, 1, n}], {n, 0, 20}]

makelist((-1)^n+sum((-1)^(n-k)*binomial(2*k-1,k-1)*(2*k)!/k!,k,1,n),n,0,12);

a(n) = Sum_{k=0..n} (-1)^(n-k)*A187535(k).
(n+2)*a(n+2) - (16*n^2 + 47*n + 34)*a(n+1) - 4*(2*n+3)^2*a(n) = 0.
a(n) ~ 2^(4*n - 1/2) * n^(n - 1/2) / (sqrt(Pi) * exp(n)). - Vaclav Kotesovec, Mar 30 2018

A187540 Binomial partial sums of the central Lah numbers.

Original entry on oeis.org

1, 3, 41, 1315, 63825, 4116611, 331127353, 31915763811, 3585520583585, 460054836028675, 66377105303195721, 10637410917472061603, 1874707445757653437681, 360356280811211873453955, 75028021167256736753934425

Offset: 0

Emanuele Munarini, Mar 11 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A187536, A008297, A111596, A187538, A187539, A187542, A187543, A187544, A187545, A187546, A187547, A187548.

seq(1+add(binomial(n,k)*binomial(2*k-1,k-1)*(2*k)!/k!, k=1..n), n=0..20);

Table[1 + Sum[Binomial[n, k]Binomial[2k-1,k-1](2k)!/k!, {k, 1, n}], {n, 0, 20}]

makelist(1+sum(binomial(n,k)*binomial(2*k-1,k-1)*(2*k)!/k!, k,1,n), n,0,12);

a(n) = 1+sum(k=0,n, binomial(n,k)*binomial(2*k-1,k-1)*(2*k)!/k!) \\ Charles R Greathouse IV, Feb 07 2017

Formula: a(n) = 1+sum(binomial(n,k)binomial(2k-1,k-1)(2k)!/k!,k=0..n).
Recurrence: for n>=3,  a(n) = 1/n*(-2 +(32 - 48*n + 16*n^2)*a(n-3) + (-31 + 63*n - 32*n^2)*a(n-2) + (3 - 14*n + 16*n^2)*a(n-1) )
E.g.f.: exp(x) (1/2 + 1/Pi K(16x) ), where K(z) is the elliptic integral of the first kind (defined as in Mathematica).
a(n) ~ 16^n*n^(n-1/2)*exp(1/16-n)/sqrt(2*Pi). - Vaclav Kotesovec, Aug 09 2013

A187542 Convolutions of the central Lah numbers (A187535).

Original entry on oeis.org

1, 4, 76, 2544, 123696, 7942080, 635633280, 61009159680, 6831940227840, 874493448514560, 125946241018214400, 20156433977646489600, 3548609812373223628800, 681555522002874494976000, 141810253720479017017344000

Offset: 0

Emanuele Munarini, Mar 11 2011

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Cf. A187536, A008297, A111596, A187536, A187538, A187539, A187540, A187543, A187544, A187545, A187546, A187547, A187548.

a := n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
seq(sum(a(k)*a(n-k), k=0..n),n=0..12);

a[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
Table[Sum[a[k]a[n - k], {k, 0, n}], {n, 0, 20}]

a(n) := if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
makelist(sum(a(k)*a(n-k),k,0,n),n,0,12);

a(n) = sum(L(k)L(n-k),k=0..n), where L(n) is a central Lah number.

a(n) ~ n! * 16^n / (Pi*n). - Vaclav Kotesovec, Oct 06 2019

Previous Showing 41-50 of 107 results. Next

The row polynomials s(n,x) := n!*L(n,4,x)= sum(a(n,m)*x^m,m=0..n) have g.f. exp(-z*x/(1-z))/(1-z)^5. They are Sheffer polynomials satisfying the binomial convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), with polynomials p(n,x)=sum(|A008297(n,m)|*(-x)^m, m=1..n) and p(0,x)=1 (for Sheffer polynomials see A048854 for S. Roman reference).

The row polynomials s(n,x) := n!*L(n,5,x)= sum(a(n,m)*x^m,m=0..n) have e.g.f. exp(-z*x/(1-z))/(1-z)^6. They are Sheffer polynomials satisfying the binomial convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), with polynomials sum(|A008297(n,m)|*(-x)^m, m=1..n), n >= 1 and p(0,x)=1 (for Sheffer polynomials see A048854 for S. Roman reference).
These polynomials appear in the radial part of the l=2 (d-wave) eigen functions for the discrete energy levels of the H-atom. See Messiah reference.
For m=0..5 the (unsigned) column sequences (without leading zeros) are: A001725(n+5), A062148-A062152. Row sums (signed) give A062191; row sums (unsigned) give A062192.
The unsigned version of this triangle is the triangle of unsigned 3-Lah numbers A143498. - Peter Bala, Aug 25 2008

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A062140 Coefficient triangle of generalized Laguerre polynomials n!*L(n,4,x) (rising powers of x).

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A062138 Coefficient triangle of generalized Laguerre polynomials n!*L(n,5,x)(rising powers of x).

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A046089 Triangle read by rows, the Bell transform of (n+2)!/2 without column 0.

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A049352 A triangle of numbers related to triangle A030524.

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A049403 A triangle of numbers related to triangle A030528; array a(n,m), read by rows (1 <= m <= n).

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A052871 Expansion of e.g.f. -LambertW(x/(-1+x)).

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